dlahrd(3S)

NAME
     DLAHRD - reduce the first NB columns of a real general n-by-(n-k+1)
     matrix A so that elements below the k-th subdiagonal are zero

SYNOPSIS
     SUBROUTINE DLAHRD( N, K, NB, A, LDA, TAU, T, LDT, Y, LDY )

         INTEGER        K, LDA, LDT, LDY, N, NB

         DOUBLE         PRECISION A( LDA, * ), T( LDT, NB ), TAU( NB ), Y(
                        LDY, NB )

IMPLEMENTATION
     These routines are part of the SCSL Scientific Library and can be loaded
     using either the -lscs or the -lscs_mp option.  The -lscs_mp option
     directs the linker to use the multi-processor version of the library.

     When linking to SCSL with -lscs or -lscs_mp, the default integer size is
     4 bytes (32 bits). Another version of SCSL is available in which integers
     are 8 bytes (64 bits).  This version allows the user access to larger
     memory sizes and helps when porting legacy Cray codes.  It can be loaded
     by using the -lscs_i8 option or the -lscs_i8_mp option. A program may use
     only one of the two versions; 4-byte integer and 8-byte integer library
     calls cannot be mixed.

PURPOSE
     DLAHRD reduces the first NB columns of a real general n-by-(n-k+1) matrix
     A so that elements below the k-th subdiagonal are zero. The reduction is
     performed by an orthogonal similarity transformation Q' * A * Q. The
     routine returns the matrices V and T which determine Q as a block
     reflector I - V*T*V', and also the matrix Y = A * V * T.

     This is an auxiliary routine called by DGEHRD.


ARGUMENTS
     N       (input) INTEGER
             The order of the matrix A.

     K       (input) INTEGER
             The offset for the reduction. Elements below the k-th subdiagonal
             in the first NB columns are reduced to zero.

     NB      (input) INTEGER
             The number of columns to be reduced.

     A       (input/output) DOUBLE PRECISION array, dimension (LDA,N-K+1)
             On entry, the n-by-(n-k+1) general matrix A.  On exit, the
             elements on and above the k-th subdiagonal in the first NB
             columns are overwritten with the corresponding elements of the
             reduced matrix; the elements below the k-th subdiagonal, with the
             array TAU, represent the matrix Q as a product of elementary
             reflectors. The other columns of A are unchanged. See Further
             Details.  LDA     (input) INTEGER The leading dimension of the
             array A.  LDA >= max(1,N).

     TAU     (output) DOUBLE PRECISION array, dimension (NB)
             The scalar factors of the elementary reflectors. See Further
             Details.

     T       (output) DOUBLE PRECISION array, dimension (LDT,NB)
             The upper triangular matrix T.

     LDT     (input) INTEGER
             The leading dimension of the array T.  LDT >= NB.

     Y       (output) DOUBLE PRECISION array, dimension (LDY,NB)
             The n-by-nb matrix Y.

     LDY     (input) INTEGER
             The leading dimension of the array Y. LDY >= N.

FURTHER DETAILS
     The matrix Q is represented as a product of nb elementary reflectors

        Q = H(1) H(2) . . . H(nb).

     Each H(i) has the form

        H(i) = I - tau * v * v'

     where tau is a real scalar, and v is a real vector with
     v(1:i+k-1) = 0, v(i+k) = 1; v(i+k+1:n) is stored on exit in A(i+k+1:n,i),
     and tau in TAU(i).

     The elements of the vectors v together form the (n-k+1)-by-nb matrix V
     which is needed, with T and Y, to apply the transformation to the
     unreduced part of the matrix, using an update of the form:  A := (I -
     V*T*V') * (A - Y*V').

     The contents of A on exit are illustrated by the following example with n
     = 7, k = 3 and nb = 2:

        ( a   h   a   a   a )
        ( a   h   a   a   a )
        ( a   h   a   a   a )
        ( h   h   a   a   a )
        ( v1  h   a   a   a )
        ( v1  v2  a   a   a )
        ( v1  v2  a   a   a )

     where a denotes an element of the original matrix A, h denotes a modified
     element of the upper Hessenberg matrix H, and vi denotes an element of
     the vector defining H(i).
SEE ALSO
     INTRO_LAPACK(3S), INTRO_SCSL(3S)

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